How to Apply Inverse Operations in Maths Problems
Inverse operations are operations that have opposing or contrary results. The result that we get can be verified by using the inverse operations. Operations like addition, subtraction, division and multiplication have their inverse operations. The inverse operation of subtraction is addition and the inverse operation of division is multiplication.
What is the Inverse of Division?
The inverse property of division is multiplication. The division consists of a dividend, a divisor and a quotient. If you multiply the quotient with the divisor, you will get the dividend as the answer.
For example, \[15 \div 3 = 5\]. Here, if we multiply 5 and 3 by each other, we will get the result of 15. This shows that the inverse of division is multiplication.
What is the Inverse of Multiplication?
The inverse of multiplication is division. If we multiply the two numbers, then the result can be used for the division as the inverse of multiplication.
For example, \[4 \times 5 = 20\] here if we divide 20 by either 4 or 5, the result we will get would be 5 or 4, respectively. This shows that the inverse of multiplication is division.
What is the Inverse of Subtraction?
The inverse of subtraction is addition. The result when added to one of the numbers will result in the other number.
For example, \[30 - 20 = 10\] here if you add 20 to 10, then it will result in 30. So, this proves that addition is the inverse of subtraction.
What is the Inverse of Addition?
The inverse of addition is subtraction. The result can be subtracted from the number and it will give us the other number.
For example, \[12 + 15 = 27\] if we subtract 27 from 15, then we will get an answer of 12. This shows that subtraction is the inverse of addition.
Inverse Operation Characteristics
Property of Inverse Addition: The additive inverse is the value that, when added to the original integer, yields 0.
Property of Inverse Multiplication: The multiplicative inverse is the value that, when multiplied by the original integer, yields 1.
Example of Additive Inverse Property
If x is the original integer, then its additive inverse is negative x, i.e., -x. If -x is the starting point, then the additive inverse will be the positive value of x, i.e., x. For example, the additive inverse of -10 would be 10 and the additive inverse of 8 would be -8.
Example of Multiplicative Inverse Property
If x is the original integer, then its multiplicative inverse is \[\dfrac{1}{x}\] and if the original integer is \[\dfrac{1}{x}\], then its multiplicative inverse will be x. If the original integer and its multiplicative are multiplied, then the result would also be 1.
For example, the multiplicative inverse of 5 would be \[\dfrac{1}{5}\] and if we multiply these two, then it would result in 1 as the answer.
Sample Questions
1. The inverse operations of \[12 + 56 = 68\] would be
a. \[68 - 56\]
b. \[68 - 12\]
c. none of the above
d. A and B
Ans: A and B
Explanation: The inverse of addition is subtraction. So, the inverse of \[12 + 56 = 68\] would be \[68 - 56\] and \[68 - 12\].
2. The inverse of multiplication is
a. addition
b. subtraction
c. division
d. all of the above
Ans: Division
3. The inverse of operation is basically
a. opposite
b. same
c. exact
d. none of the above
Ans: Opposite
Conclusion
The inverse operation will change from addition to subtraction and from division to multiplication and vice versa. The inverse of multiplication when multiplied with the result gives 1 as the answer. The inverse operations help in verifying the answers.
FAQs on Inverse Operations Explained for Students
1. What are inverse operations in simple terms?
In mathematics, inverse operations are pairs of operations that undo each other. If you start with a number, perform an operation, and then perform its inverse operation, you will get back to the original number. For example, if you add 5 to a number, subtracting 5 will reverse that action.
2. What are the main pairs of inverse operations in Maths?
The most common pairs of inverse operations that students learn in the CBSE/NCERT syllabus are:
- Addition and Subtraction: Adding a number is undone by subtracting that same number.
- Multiplication and Division: Multiplying by a number is undone by dividing by that same number.
- Squaring and Square Root: Finding the square of a number (e.g., 4²) is undone by taking its square root (√16).
3. How are addition and subtraction related as inverse operations?
Addition and subtraction are directly opposite. They form a fact family. For example, if we know that 7 + 3 = 10, we can use the inverse operation (subtraction) to create two related facts: 10 - 3 = 7 and 10 - 7 = 3. This relationship helps in checking answers and solving equations.
4. How do multiplication and division work as inverse operations?
Multiplication and division are inverse operations because one reverses the effect of the other. If you multiply a number, you can get back to the original number by dividing. For instance, if you calculate 6 × 5 = 30, the inverse operation would be to divide the result by 5: 30 ÷ 5 = 6, which brings you back to the original number.
5. How are inverse operations different from additive or multiplicative inverses?
This is a common point of confusion. Inverse operations are actions that reverse each other (e.g., addition and subtraction). In contrast, additive and multiplicative inverses (also called reciprocals) are specific numbers. The additive inverse of a number 'a' is '-a' because a + (-a) = 0. The multiplicative inverse of 'a' is '1/a' because a × (1/a) = 1. So, one is about undoing an action, while the other is about finding a number that gives a neutral result (0 or 1).
6. Can you provide a real-life example of inverse operations?
A simple real-life example is operating a light switch. Flipping the switch ON is an operation. To undo this, you perform the inverse operation: flipping the switch OFF. Another example is closing a door (operation) and opening it (inverse operation) to return it to its original state.
7. How do inverse operations help in checking answers to math problems?
Inverse operations are a powerful tool for verification. After solving a problem, you can use the inverse operation to work backwards from your answer. For example, if you solved the subtraction problem 25 - 10 = 15, you can check your work by performing the inverse operation (addition): 15 + 10 = 25. Since you got back to the original number, your answer is correct.
8. What is the inverse operation of squaring a number?
The inverse operation of squaring a number is finding its square root. For example, if you take the number 9 and square it, you get 9² = 81. To undo this, you find the square root of 81, which is √81 = 9. This brings you back to your starting number.
9. Why are inverse operations so important for solving algebraic equations?
Inverse operations are the foundation of solving algebraic equations. The goal in algebra is often to isolate a variable (like 'x'). To do this, you must undo all the operations being performed on that variable. For example, to solve the equation x + 8 = 15, you must undo the 'add 8' operation by applying its inverse: subtracting 8 from both sides of the equation.
10. Do inverse operations always work perfectly?
For basic arithmetic, yes. However, in more advanced topics, there can be nuances. For example, the inverse of squaring a number is taking the square root. But if you start with a negative number like -4, square it to get 16, and then take the principal square root (√16), you get +4, not the original -4. Similarly, you cannot use division by zero as an inverse to multiplication by zero. Understanding these limitations is important as you advance in mathematics.
